Dual-Path State Reconstruction Scheme for Propagating Quantum Microwaves and Detector Noise Tomography

Quantum state reconstruction of weak propagating microwaves to date requires the use of linear amplifiers. We introduce a theory which, even in the presence of significant amplifier noise, allows one to use these devices for measuring all quadrature moments of propagating quantum microwaves based on cross correlations from a dual-path amplification setup. Simultaneously, the detector noise properties are determined, allowing for tomography. We demonstrate the feasibility of our novel concept by proof-of-principle experiments with classical mixtures of weak coherent microwaves.

Figure 1

(a) Schematics of the dual-path amplification and detection setup. The ancilla port of the 180° hybrid ring is terminated by a $50\Omega$ load at $T=300\mathrm{mK}$. The split signals are amplified and detected in separate chains (rounded triangular symbols), which add the noise ${\chi }_1$ and ${\chi }_2$ to the signal. (b) Amplification and detection chain. The noise temperature is dominated by the cryogenic amplifiers anchored at 4.2 K. The isolation of the hybrid ring (40 dB) and the circulator prevents correlated noise contributions from the two chains. The phase difference between $C_1$ and $C_2$ is close to 180°.

Figure 2

Detection of coherent microwave probing pulses (${10}^7$ traces averaged). (a) Mean value of down converted signal (10 MHz). (b) Autovariance (blue) and cross variance (red). (c) Autovariance (blue) and cross variance (red) fluctuations, obtained by subtracting the respective time averages. (d) Dynamic range of the mean value. $P_{\mathrm{rms}}$ is the root mean square power at the digitizer. The numbers above the data points indicate the number of photons on average. (e) Dynamic range of the cross product $⟨C_1{C}_2⟩$. The arrows indicate the values obtained from the time traces shown in (f). The rms amplitude inside the pulse region is calculated after subtracting the time average from each data point. (f) Power dependence of the cross product time traces in the limit of few photons on average. (g) Discrete Fourier transform (DFT) of the pulse region of the traces shown in (f). The arrows indicate the 20 MHz peaks.

Figure 3

Statistical mixtures of phase-shifted pulses. (a) Cross variance amplitude $\alpha$ plotted versus $\phi$ at 2 POA. Solid line: fit to the data. Inset: Pulse distribution histogram. (b) Third central moment at 100 POA measured with a single amplification chain. As expected, the trace is $\propto {sin}^3(\omega t)$. Insets: Equally distributed (brown) and finite skew (green) histograms. (c) Third central moment measured with the dual-path method, data set as in (c). (d) Power dependence of the third central moment. Gray area: detection limit (10–20 POA). For technical reasons, in (a) and (d) a Wilkinson power divider [17, 18] was used instead of the hybrid ring.